Frequency response of an LTI system

CTFT of the impulse response is the system’s frequency response, and is denoted by $H(j \omega) = \int _{\infty} ^{\infty} h(t) e^{-j\omega t}dt$. An LTI system amplitude-scales and phase-shifts any complex exponential input!

• “Eigenfunction” property: if $x(t) = e^{j \omega _0 t}$, then $y(t) \propto e^{j \omega_0 t}$
• Amplitude scaling: output amplitude is scaled by $|H(j \omega _0)|$
• Phase shifting: output phase is shifted by $\angle H(j \omega _0)$

Response of an LTI system and the CTFT

In the frequency domain, the output spectrum $Y(j \omega)$ is given by the product of the frequency response $H(j \omega)$ and the input spectrum $X(j \omega)$

LTI systems and invertibility

An LTI system $T$ with impulse response $h$ is invertible if and only if $H(j\omega)≠0$ for all $\omega \in \R$

• Negative feedback combination of two LTI systems is the LTI system $T$ with impulse response

  

Filtering via LTI systems

Filters can be conveniently designed using LTI systems through an appropriately chosen frequency response

Ideal low-pass filter

• If the frequency response $H(j \omega)$ is the unit step function between $+/- \omega _c$, the impulse response is $h(t) = \frac{\sin(\omega _c t)}{\pi t}$

  

  • This filter is not causal since $h(t)$ is not zero for all $t < 0$, so we cannot use this ideal filter in any real-time application
  • Ideal LPF impulse response oscillates when convolved in the time-domain, this tends to produce undesirable oscillations in the output signal

• We prefer non-ideal but causal filters

  • They can be modelled using differential equations

Ideal high-pass filter

• If the frequency response $H(j \omega)$ is the unit step function between $+/- \omega _c$, the impulse response is $h(t) = \delta (t) = - \frac{\sin(\omega _c t)}{\pi t}$